Theorem ssuzfz 37659

Description: A finite subset of the upper integers is a subset of a finite set of sequential integers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
ssuzfz.1	|- Z = ( ZZ>= ` M )
ssuzfz.2	|- ( ph -> A C_ Z )
ssuzfz.3	|- ( ph -> A e. Fin )

Assertion

Ref	Expression
ssuzfz	|- ( ph -> A C_ ( M ... sup ( A , RR , < ) ) )

Proof of Theorem ssuzfz

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssuzfz.2	. . . . . . . . . 10 |- ( ph -> A C_ Z )
2	1	sselda 3418	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> k e. Z )
3		ssuzfz.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
4	2, 3	syl6eleq 2559	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> k e. ( ZZ>= ` M ) )
5		eluzel2 11187	. . . . . . . 8 |- ( k e. ( ZZ>= ` M ) -> M e. ZZ )
6	4, 5	syl 17	. . . . . . 7 |- ( ( ph /\ k e. A ) -> M e. ZZ )
7		uzssz 11202	. . . . . . . . . . . 12 |- ( ZZ>= ` M ) C_ ZZ
8	3, 7	eqsstri 3448	. . . . . . . . . . 11 |- Z C_ ZZ
9	8	a1i 11	. . . . . . . . . 10 |- ( ph -> Z C_ ZZ )
10	1, 9	sstrd 3428	. . . . . . . . 9 |- ( ph -> A C_ ZZ )
11	10	adantr 472	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> A C_ ZZ )
12		ne0i 3728	. . . . . . . . . 10 |- ( k e. A -> A =/= (/) )
13	12	adantl 473	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> A =/= (/) )
14		ssuzfz.3	. . . . . . . . . 10 |- ( ph -> A e. Fin )
15	14	adantr 472	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> A e. Fin )
16		suprfinzcl 11073	. . . . . . . . 9 |- ( ( A C_ ZZ /\ A =/= (/) /\ A e. Fin ) -> sup ( A , RR , < ) e. A )
17	11, 13, 15, 16	syl3anc 1292	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> sup ( A , RR , < ) e. A )
18	11, 17	sseldd 3419	. . . . . . 7 |- ( ( ph /\ k e. A ) -> sup ( A , RR , < ) e. ZZ )
19	10	sselda 3418	. . . . . . 7 |- ( ( ph /\ k e. A ) -> k e. ZZ )
20	6, 18, 19	3jca 1210	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ k e. ZZ ) )
21		eluzle 11195	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> M <_ k )
22	4, 21	syl 17	. . . . . 6 |- ( ( ph /\ k e. A ) -> M <_ k )
23		zssre 10968	. . . . . . . . . 10 |- ZZ C_ RR
24	23	a1i 11	. . . . . . . . 9 |- ( ph -> ZZ C_ RR )
25	10, 24	sstrd 3428	. . . . . . . 8 |- ( ph -> A C_ RR )
26	25	adantr 472	. . . . . . 7 |- ( ( ph /\ k e. A ) -> A C_ RR )
27		simpr 468	. . . . . . 7 |- ( ( ph /\ k e. A ) -> k e. A )
28		eqidd 2472	. . . . . . 7 |- ( ( ph /\ k e. A ) -> sup ( A , RR , < ) = sup ( A , RR , < ) )
29	26, 15, 27, 28	supfirege 10620	. . . . . 6 |- ( ( ph /\ k e. A ) -> k <_ sup ( A , RR , < ) )
30	20, 22, 29	jca32 544	. . . . 5 |- ( ( ph /\ k e. A ) -> ( ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ k e. ZZ ) /\ ( M <_ k /\ k <_ sup ( A , RR , < ) ) ) )
31		elfz2 11817	. . . . 5 |- ( k e. ( M ... sup ( A , RR , < ) ) <-> ( ( M e. ZZ /\ sup ( A , RR , < ) e. ZZ /\ k e. ZZ ) /\ ( M <_ k /\ k <_ sup ( A , RR , < ) ) ) )
32	30, 31	sylibr 217	. . . 4 |- ( ( ph /\ k e. A ) -> k e. ( M ... sup ( A , RR , < ) ) )
33	32	ex 441	. . 3 |- ( ph -> ( k e. A -> k e. ( M ... sup ( A , RR , < ) ) ) )
34	33	ralrimiv 2808	. 2 |- ( ph -> A. k e. A k e. ( M ... sup ( A , RR , < ) ) )
35		dfss3 3408	. 2 |- ( A C_ ( M ... sup ( A , RR , < ) ) <-> A. k e. A k e. ( M ... sup ( A , RR , < ) ) )
36	34, 35	sylibr 217	1 |- ( ph -> A C_ ( M ... sup ( A , RR , < ) ) )

Syntax hints:   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   C_ wss 3390   (/)c0 3722   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   RRcr 9556   < clt 9693   <_ cle 9694   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811

This theorem is referenced by:  sge0isum  38383